Description:

An accessible introduction to the basic elements of algebraic codes including Reed-Solomon, trellis, turbocodes, etc. Throughout the book, mathematical theory is illustrated by reference to many practical examples.

Richard E. Blahut is Head of the Department of Electrical and Computer Engineering at the University of Illinois, Urbana Champaign, where he is also a professor. He is a Fellow of the IEEE and the recipient of many awards including the IEEE Alexander Graham Bell Medal (1998), the Tau Beta Pi Daniel C. Drucker Eminent Faculty Award, and the IEEE Millennium Medal. He was named Fellow of the IBM Corporation in 1980, where he worked for over 30 years, and was elected to the National Academy of Engineering in 1990.

Preface

Introduction

The discrete communication channel

The history of data-transmission codes

Applications

Elementary concepts

Elementary codes

Problems

Introduction to Algebra

Fields of characteristic two

Groups

Rings

Fields

Vector spaces

Linear algebra

Problems

Notes

Linear Block Codes

Structure of linear block codes

Matrix description of linear block codes

Hamming codes

The standard array

Hamming spheres and perfect codes

Simple modifications to a linear code

Problems

Notes

The Arithmetic of Galois Fields

The integer ring

Finite fields based on the integer ring

Polynomial rings

Finite fields based on polynomial rings

Primitive elements

The structure of finite fields

Problems

Notes

Cyclic Codes

Viewing a code from an extension field

Polynomial description of cyclic codes

Minimal polynomials and conjugates

Matrix description of cyclic codes

Hamming codes as cyclic codes

Cyclic codes for correcting double errors

Quasi-cyclic codes and shortened cyclic codes

The Golay code as a cyclic code

Cyclic codes for correcting burst errors

The Fire codes as cyclic codes

Cyclic codes for error detection

Problems

Notes

Codes Based on the Fourier Transform

The Fourier transform

Reed-Solomon codes

Conjugacy constraints and idempotents

Spectral description of cyclic codes

BCH codes

The Peterson-Gorenstein-Zierler decoder

The Reed-Muller codes as cyclic codes

Extended Reed-Solomon codes

Extended BCH codes

Problems

Notes

Algorithms Based on the Fourier Transform

Spectral estimation in a finite field

Synthesis of linear recursions

Decoding of binary BCH codes

Decoding of nonbinary BCH codes

Decoding with erasures and errors

Decoding in the time domain

Decoding within the BCH bound

Decoding beyond the BCH bound

Decoding of extended Reed-Solomon codes

Decoding with the euclidean algorithm

Problems

Notes

Implementation

Logic circuits for finite-field arithmetic

Shift-register encoders and decoders

The Meggitt decoder

Error trapping

Modified error trapping

Architecture of Reed-Solomon decoders

Multipliers and inverters

Bit-serial multipliers

Problems

Notes

Convolutional Codes

Codes without a block structure

Trellis description of convolutional codes

Polynomial description of convolutional codes

Check matrices and inverse matrices

Error correction and distance notions

Matrix description of convolutional codes

The Wyner-Ash codes as convolutional codes

Syndrome decoding algorithms

Convolutional codes for correcting error bursts

Algebraic structure of convolutional codes

Problems

Notes

Beyond BCH Codes

Product codes and interleaved codes

Bicyclic codes

Concatenated codes

Cross-interleaved codes

Turbo codes

Justesen codes

Problems

Notes

Codes and Algorithms Based on Graphs

Distance, probability, and likelihood

The Viterbi algorithm

Sequential algorithms to search a trellis

Trellis description of linear block codes

Gallager codes

Tanner graphs and factor graphs

Posterior probabilities

The two-way algorithm

Iterative decoding of turbo codes

Tail-biting representations of block codes

The Golay code as a tail-biting code

Problems

Notes

Performance of Error-Control Codes

Weight distributions of block codes

Performance of block codes

Bounds on minimum distance of block codes

Binary expansions of Reed-Solomon codes

Symbol error rates on a gaussian-noise channel

Sequence error rates on a gaussian-noise channel

Coding gain

Capacity of a gaussian-noise channel

Problems

Notes

Codes and Algorithms for Majority Decoding

Reed-Muller codes

Decoding by majority vote

Circuits for majority decoding

Affine permutations for cyclic codes

Cyclic codes based on permutations

Convolutional codes for majority decoding

Generalized Reed-Muller codes

Euclidean-geometry codes

Projective-geometry codes

Problems

Notes

Bibliography

Index

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